At Crypto 2012, Brakerski constructed a scale-invariant fully homomorphic encryption scheme based on the LWE problem, in which the same modulus is used throughout the evaluation process, instead of a ladder of moduli when doing "modulus switching". In this paper we describe a variant of the van Dijk et al. FHE scheme over the integers with the same scale-invariant property. Our scheme has a single secret modulus whose size is linear in the multiplicative depth of the circuit to be homomorphically evaluated, instead of exponential; we therefore construct a leveled fully homomorphic encryption scheme. This scheme can be transformed into a pure fully homomorphic encryption scheme using bootstrapping, and its security is still based on the Approximate-GCD problem. We also describe an implementation of the homomorphic evaluation of the full AES encryption circuit, and obtain significantly improved performance compared to previous implementations: about 23 seconds (resp. 3 minutes) per AES block at the 72-bit (resp. 80-bit) security level on a mid-range workstation. Finally, we prove the equivalence between the (error-free) decisional Approximate-GCD problem introduced by Cheon et al. (Eurocrypt 2013) and the classical computational Approximate-GCD problem. This equivalence allows to get rid of the additional noise in all the integer-based FHE schemes described so far, and therefore to simplify their security proof. © 2014 Springer-Verlag Berlin Heidelberg.
CITATION STYLE
Coron, J. S., Lepoint, T., & Tibouchi, M. (2014). Scale-invariant fully homomorphic encryption over the integers. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 8383 LNCS, pp. 311–328). Springer Verlag. https://doi.org/10.1007/978-3-642-54631-0_18
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